Abstract
In this work, we study the problem of learning a partial differential equation (PDE) from its solution data. PDEs of various types are used to illustrate how much the solution data can reveal the PDE operator depending on the underlying operator and initial data. A data-driven and data-adaptive approach based on local regression and global consistency is proposed for stable PDE identification. Numerical experiments are provided to verify our analysis and demonstrate the performance of the proposed algorithms.
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Acknowledgements
H. Zhao’s research is partially supported by NSF Grant DMS-2012860 and DMS-2309551. Y. Zhong’s research is partially supported by NSF Grant DMS-2309530.
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Appendix A. Proof of (4.20) and (4.21)
Appendix A. Proof of (4.20) and (4.21)
In the following, we assume \(N > 1\). Recalling that the variance of a random variable X can be expressed as \({\mathbb {E}}[X^2]-({\mathbb {E}}[X])^2\), we get
Hence by defining the estimator:
and noting the Lipschitz assumption, we have
As for the variance of the estimator, we notice that since \(\zeta _n\), \(n=1,\dots , N\) are independent Gaussian random variables, if we denote \(S:=\sqrt{\frac{B-1}{B}}\sigma \), \(\sum _{n=1}^N\zeta _n^2/S^2\) has a non-central Chi-squared distribution whose mean is \(N+\sum _{n=1}^N\mu _n^2/S^2\), and variance is \(2(N+2\sum _{n=1}^N\mu _n^2/S^2)\); and \((\sum _{n=1}^N\zeta _n)^2/(NS^2)\) also has a non-central Chi-squared distribution whose mean is \(1+(\sum _{n=1}^N\mu _n)^2/(NS^2)\), and variance is \(2(1+2(\sum _{n=1}^N\mu _n)^2/(NS^2))\). First, we compute the covariance
Focusing on the first term, we have
Hence, we have
Now we note that
After simplification, we get
Considering the Lipschitz assumption, we obtain
Therefore, denoting \(\gamma =4DL^2R^2\), then we get
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He, Y., Zhao, H. & Zhong, Y. How Much Can One Learn a Partial Differential Equation from Its Solution?. Found Comput Math (2023). https://doi.org/10.1007/s10208-023-09620-z
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DOI: https://doi.org/10.1007/s10208-023-09620-z